MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nb3grprlem1 Structured version   Visualization version   GIF version

Theorem nb3grprlem1 27089
Description: Lemma 1 for nb3grpr 27091. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
nb3grpr.v 𝑉 = (Vtx‘𝐺)
nb3grpr.e 𝐸 = (Edg‘𝐺)
nb3grpr.g (𝜑𝐺 ∈ USGraph)
nb3grpr.t (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
Assertion
Ref Expression
nb3grprlem1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Proof of Theorem nb3grprlem1
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nb3grpr.s . . . . . . 7 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
2 prid1g 4688 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐵, 𝐶})
323ad2ant2 1126 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐵 ∈ {𝐵, 𝐶})
41, 3syl 17 . . . . . 6 (𝜑𝐵 ∈ {𝐵, 𝐶})
54adantr 481 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
6 eleq2 2898 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
76eqcoms 2826 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
87adantl 482 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
95, 8mpbid 233 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (𝐺 NeighbVtx 𝐴))
10 nb3grpr.g . . . . . 6 (𝜑𝐺 ∈ USGraph)
11 nb3grpr.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
1211nbusgreledg 27062 . . . . . . 7 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13 prcom 4660 . . . . . . . . 9 {𝐵, 𝐴} = {𝐴, 𝐵}
1413a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
1514eleq1d 2894 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
1612, 15bitrd 280 . . . . . 6 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1710, 16syl 17 . . . . 5 (𝜑 → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1817adantr 481 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
199, 18mpbid 233 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20 prid2g 4689 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐵, 𝐶})
21203ad2ant3 1127 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐵, 𝐶})
221, 21syl 17 . . . . . 6 (𝜑𝐶 ∈ {𝐵, 𝐶})
2322adantr 481 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
24 eleq2 2898 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2524eqcoms 2826 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2625adantl 482 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2723, 26mpbid 233 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
2811nbusgreledg 27062 . . . . . . 7 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29 prcom 4660 . . . . . . . . 9 {𝐶, 𝐴} = {𝐴, 𝐶}
3029a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
3130eleq1d 2894 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
3228, 31bitrd 280 . . . . . 6 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3310, 32syl 17 . . . . 5 (𝜑 → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3433adantr 481 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3527, 34mpbid 233 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ 𝐸)
3619, 35jca 512 . 2 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
37 nb3grpr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
3837, 11nbusgr 27058 . . . . 5 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
3910, 38syl 17 . . . 4 (𝜑 → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
4039adantr 481 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
41 nb3grpr.t . . . . . . . . . 10 (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
42 eleq2 2898 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4341, 42syl 17 . . . . . . . . 9 (𝜑 → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4443adantr 481 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
45 vex 3495 . . . . . . . . . . 11 𝑣 ∈ V
4645eltp 4618 . . . . . . . . . 10 (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶))
4711usgredgne 26915 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴𝑣)
48 df-ne 3014 . . . . . . . . . . . . . . . . 17 (𝐴𝑣 ↔ ¬ 𝐴 = 𝑣)
49 pm2.24 124 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5049eqcoms 2826 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5150com12 32 . . . . . . . . . . . . . . . . 17 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5248, 51sylbi 218 . . . . . . . . . . . . . . . 16 (𝐴𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5347, 52syl 17 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5453ex 413 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5510, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5655adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5756com3r 87 . . . . . . . . . . 11 (𝑣 = 𝐴 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
58 orc 861 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣 = 𝐵𝑣 = 𝐶))
59582a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
60 olc 862 . . . . . . . . . . . 12 (𝑣 = 𝐶 → (𝑣 = 𝐵𝑣 = 𝐶))
61602a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6257, 59, 613jaoi 1419 . . . . . . . . . 10 ((𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6346, 62sylbi 218 . . . . . . . . 9 (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6463com12 32 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6544, 64sylbid 241 . . . . . . 7 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6665impd 411 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐵𝑣 = 𝐶)))
67 eqid 2818 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
68673mix2i 1326 . . . . . . . . . . . . . . . . 17 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
691simp2d 1135 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑌)
70 eltpg 4615 . . . . . . . . . . . . . . . . . 18 (𝐵𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7169, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7268, 71mpbiri 259 . . . . . . . . . . . . . . . 16 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
7372adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74 eleq1 2897 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
7574bicomd 224 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7675adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7773, 76mpbid 233 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
7842bicomd 224 . . . . . . . . . . . . . . . 16 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
7941, 78syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8079adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8177, 80mpbid 233 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐵) → 𝑣𝑉)
8281ex 413 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐵𝑣𝑉))
8382adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵𝑣𝑉))
8483impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
85 preq2 4662 . . . . . . . . . . . . . . 15 (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
8685eleq1d 2894 . . . . . . . . . . . . . 14 (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8786eqcoms 2826 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8887biimpcd 250 . . . . . . . . . . . 12 ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
8988ad2antrl 724 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
9089impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
9184, 90jca 512 . . . . . . . . 9 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
9291ex 413 . . . . . . . 8 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
93 tpid3g 4700 . . . . . . . . . . . . . . . . . 18 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
94933ad2ant3 1127 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
951, 94syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
9695adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
97 eleq1 2897 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9897bicomd 224 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
9998adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
10096, 99mpbid 233 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
10179adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
102100, 101mpbid 233 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐶) → 𝑣𝑉)
103102ex 413 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐶𝑣𝑉))
104103adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶𝑣𝑉))
105104impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
106 preq2 4662 . . . . . . . . . . . . . . 15 (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107106eleq1d 2894 . . . . . . . . . . . . . 14 (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108107eqcoms 2826 . . . . . . . . . . . . 13 (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109108biimpcd 250 . . . . . . . . . . . 12 ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110109ad2antll 725 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
111110impcom 408 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
112105, 111jca 512 . . . . . . . . 9 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
113112ex 413 . . . . . . . 8 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11492, 113jaoi 851 . . . . . . 7 ((𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
115114com12 32 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 = 𝐵𝑣 = 𝐶) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11666, 115impbid 213 . . . . 5 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵𝑣 = 𝐶)))
117116abbidv 2882 . . . 4 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)})
118 df-rab 3144 . . . 4 {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119 dfpr2 4576 . . . 4 {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)}
120117, 118, 1193eqtr4g 2878 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
12140, 120eqtrd 2853 . 2 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
12236, 121impbida 797 1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3o 1078  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wne 3013  {crab 3139  {cpr 4559  {ctp 4561  cfv 6348  (class class class)co 7145  Vtxcvtx 26708  Edgcedg 26759  USGraphcusgr 26861   NeighbVtx cnbgr 27041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-edg 26760  df-upgr 26794  df-umgr 26795  df-usgr 26863  df-nbgr 27042
This theorem is referenced by:  nb3grpr  27091  nb3grpr2  27092  nb3gr2nb  27093
  Copyright terms: Public domain W3C validator